A ply drop-off element for inclusion of drop-off in the global analysis of layered composite structures

Pergamon

0045-7949(94)00331-9

Compurerr & Slrurrurc.r Vol. 54. No. 5. pp. 865-870. 1995 Elsevier Science Lrd Printed in Great Britain 0045-7949/95 119 50 + 0.00

A PLY DROP-OFF ELEMENT FOR INCLUSION OF DROP-OFF IN THE GLOBAL ANALYSIS OF LAYERED COMPOSITE STRUCTURES and B. Varughese

A. Mukherjeet Department

of Civil Engineering,

Indian

Institute

of Technology,

(Received 4 September

Bombay

400 076, India

1993)

Abstract-A finite element approach for the inclusion of the ply drop-off in the global analysis of structures made of layered composite materials has been presented. The proposed element can accommodate the drop-off anywhere within the element. Therefore, it can include the effect of drop-off without increasing the size of the global matrices. The applicability of the method has been illustrated by means of several examples.

be impracticably small. A coarser mesh, however, cannot incorporate the drop-off accurately (Fig. 2). Therefore, the study of ply drop-off has been restricted to the local analysis of simple structures only and the authors have been unable to locate any attempt to include the drop-off in the global model of practical layered composite structures. This investigation aims to develop a ply drop-off element which can accommodate a drop-off anywhere within the element and the drop-off need not pass through a nodal line. Therefore, the mesh division can be independent of the location of drop-offs. This element can be used in the global model without increasing the size of the global structural matrices. For accurate prediction of the stress field in the vicinity of the drop-off, however, a local analysis using a finer mesh needs to be undertaken. The results of the global analysis can be used as the input for the local analysis. This study concentrates on the drop-off effects in wing or fin skin structures. Such structures are analysed by considering only the membrane effect. Therefore, in this investigation, only membrane effects due to drop-off are considered. As a first step, only beams with a step have been analysed using the proposed approach. This can, however, be extended

1. lNTRODUCTION

A ply drop refers to a change in laminate thickness as a result of terminating internal plies or as a result of inserting internal plies (Fig. 1). Wing structures of aircraft made of composite laminates may have dropped plies to help the wing to have thicker sections at its roots and thinner sections away from it. Also, plies may be inserted at access holes and joints to act as stiffeners. Although ply drop studies have been carried out for a long time, very few unclassified documents are available. Kemp and Johnson [I] conducted a finite element analysis to study the stress distributions at the vicinity of the ply drop. From their studies it can be seen that ply drop introduces a few additional modes of failure. The stress concentration due to the drop-off may lead to debonding. The weak resin pocket generated due to the drop-off may fail as well. Curry, Johnson and Starnes [2] conducted experimental, as well as analytical, studies to observe the effect of the number of dropped plies and their orientation on the laminate’s compressive strength. It was found that the reduction in axial strength is directly related to the axial stiffness change between the thick and thin sections. The loss of strength due to ply drop-off can be reduced significantly by introducing the dropoff gradually. Therefore, a few rules are followed for dropping the plies off. Plies should be dropped off with a stagger of at least 6 mm. Dropped plies are interleaved with continuous plies. As a result, instead of forming a step at a point, the drop-off is spread over a region. Therefore, it is very difficult to accommodate the drop accurately in the global finite element model of the structure. If a nodal line is placed along every drop-off line, the sizes of the elements will

t Author to whom all correspondence

Stagger distance I-

I

I

Fig. I. A typical

is to be addressed. 865

ply drop-off.

A. Mukherjee and B. Varughese

866

xX = L, respectively. point is given by

In a layer, the coordinate

x = N,x, + N2x2 + N,x,,

I .e3I

0.5L

0.5L

Fig. 2. Integration

points and their mapping metric coordinates.

I into isopara-

The formulation has been developed using either a three-noded or a two-noded isoparametric element. The displacement at any point in the element at a distance z from the reference plane is given by, u = u f ZB, w = w,

(1)

where u and w are the displacements in the x and z directions, 0, is the average rotation of the normal about the y-axis. Using the above relationship, the stiffness matrix can be written as

where B, and D, are the strain-displacement relationship matrix and the rigidity matrix, respectively, for the ith layer and NL is the total number of layers.

3.

METHOD

OF INTEGRATION

Although eqn (2) can be integrated analytically, a Gaussian integration has been used for the integration here; this has been done with a view to extending the concept to a more general case of plates and shells. The three-noded elements require two point integration while the two-noded elements require one point integration. The integration has to be carried out for each layer separately; this is because the lengths of the dropped-off layers are not the same as that of the element length. The points of integration for each dropped-off layer have been found out by picking up the Gaussian integration points on the dropped-off layer. In Fig. 2, a three-noded beam element is shown and it is assumed that the first, second and third nodes are at x, = 0, x2 = L/2 and

(3)

where N,, N, and N, are the shape functions at the three nodes. It may be noted that the functions N,, N, and N, are quadratic of 5. Therefore, it is expected that eqn (3) should result in a nonlinear relationship. However, for a straight beam, the values 0, l/2 and 1 can be substituted for x, , .x2and x3 in eqn (3) and the final expression is a linear one. That is,

to the study of composite plates and shells with drop-off very easily. A good correlation between the proposed method and the exact analysis has been found. 2. STIFFNESS MATRIX

of any

x = (1 + 5 )L/2.

(4)

It can be seen that the domain of integration for the dropped-off layer is smaller than that of the whole element. This should reflect in the Jacobian J in eqn (2). In the present formulation, the Jacobian is calculated as L,/2, where L, is the length of the ith layer. A close look at the B matrix reveals that it contains only the first order derivatives of the shape functions. Therefore, for the three-noded elements which have quadratic shape functions, the variation in the B matrix is linear. For the two-noded elements, however, the derivative of shape function is constant. Therefore the B matrix does not reflect the location of drop-off at all. The contribution of the dropped-off layer is distributed equally among the two nodes of the element.

4. NUMERICAL

RESULTS

Figure 3 shows the geometry of the beam considered in the numerical examples. It is a three-layered beam with two outer layers being dropped-off. In order to achieve the membrane effect, the beam is loaded at the free end in the positive x-direction. The span (L) of the beam and the tip load (P) are kept at 10.0 and 5.0 respectively in all the examples. To specify the location of the drop-off in the present discussion, a term called the drop ratio (D.R.) has been introduced. The drop ratio (DA) is the ratio of the lengths (Ll) of the dropped-off layers to the span (L) of the beam.

0.25 0.50 0.25

Fig. 3. Geometry

of the layered examples.

beam

in the numerical

Drop-off

in composite

2.00

D.R.zO.1

____....~.........._.......~

structure

D..--

D.R.=0.2 _~~.~~~“~~~__.~~_~..~.....~..~

1.8I_ q-

1.t i-

D.R.=O. I

D.R.=0.5

q ‘..c

I .AI-

D.R.=0.6

____.._.~.................~

..a

0’

r

D.R.=0.4

_,___....~..__..._..........~ 0”

-0

867

D.R.=0.3

O___‘_“~.........‘........

q“-

analysis

D.R.=0.7 D.R.=0.7

D.R.=9.8

D.R.=0.8 .................

D.R.=0.9

__~....‘.“D_.“..‘.........~

1.:!-

0. ..a--------w

0.

D” IS )-

Ll

I

I

I2

a

.O

I

o--o

FEM & Exact

I

3

I 4

(d) of isotropic (example

I 5

I I 678

I

I 9

Number Fig. 6. Deflection

of elements tip loaded cantilever 1).

of elements

(d) of composite tip loaded beam (example 3).

Studies have been conducted considering the beam for the following three examples: Example 1. Isotropic. Material properties (in consistent units), E = 1.0, G = 0.38, v = 0.30. Example 2. Composite-lay-up (45/O/45). Example 3. Composite-lay-up (90/O/90). Material properties (in consistent units) for examples 2 and 3 are EL/ET = 40.O,G,,/E,

= 0.6,G,/E,

= 0.3

VLT= VTZ= 0.25.

D

4.1. Effect of mesh size for a particular

6.25 -

D.R.=0.2 q

0 d--

6.00 D.R.=0.4 a D.R.=0.5 cl D.R.=0.6

-

: 5.75 -

-

D.R.=0.8 cl 0

D.R.=l Exact D.--O 1 I I I 2 3 4 5 Number

Fig. 5. Deflection

:

‘. ‘. :

‘.

: : :

’

:

D.R.=O.l D.R.=0.2 D.R.=0.3 D.R.=0.4 D.R.=0.5 D.R.=0.6 D.RsO.7 D.R.=O.t? D.R.=0.9 D.R.= I .O &

Exact

5.25 -

D.R.=0.9

1.93 -

I

*. :

: : : q : : :

b rA 5.50-

D.R.=0.7 a

1.94 -

-

:

: :

D.R.=0.3

-

--o---o--a---+-----+--X-- -+- --o--

0

0 ’ : :

I-. P

II ‘.92o

drop ratio

To examine the present element’s capability to accommodate the drop-off anywhere within the element, the cantilever beam mentioned above has been modeled using various elements and mesh divisions. Two types of elements, namely two-noded and threenoded elements, have been used in this section. Comparison has been made between three different meshings: (1) two two-noded elements; (2) four twonoded elements; (3) two three-noded elements.

0”

1.95 -

two

1. Effect of mesh size for a particular drop ratio. 2. Effect of changing drop ratio on the response.

D.R.=O.l

1.99-

Q 1.96-

the following

6.50-

2.00-

1.9-I-

cantilever

beam

The study aims to investigate effects:

1.98-

.O rl

Exact FEM

Number Fig. 4. Deflection

D.R.=l

1.95 D.R.=I

L

D---O 0.1 gO

D.R.=O.9

~__“.~‘~~~“~~~~.“..~~~...~

5.00 -

.O

FEM II

6

I

I

I

8

9

II 4’75~

I

11 2

Number

of elements

(d) of composite tip loaded beam (example 2).

3

cantilever

Fig. 7. Stress values

11 4

5

11 6

7

1

8

I 9

of elements

before the drop-off ratios (example I).

for various

drop

A. Mukherjee

868 Table

I. Non-dimensional

deflection

d for various

Composite 45/o/45

Isotropic

meshings

--o---o---II+--+--. .-fr--X---*---o--

9.95-

Composite 90/o/90

A

B

A

B

A

B

0.500

1.500

0.964

1.964

0.975

.976

0.500

IS00

0.964

1.964

0.975

,976

0.500

1.500

0.964

1.964

0.975

,976

0.545

I.455

0.969

1.964

0.979

,975

0.539

1.460

0.969

1.964

0.979

I.976

0.532

I.467

0.967

1.964

0.978

I.975

0.583

1.417

0.974

1.964

0.982

1.975

0.560

1.440

0.970

1.964

0.979

1.975

0.543

1.457

0.968

1.964

0.978

0.615

1.385

0.978

1.964

0.985

0.560

I.440

0.970

1.964

0.979

0.544

1.456

0.968

1.964

0.978

0.643

1.357

0.980

1.963

0.986

0.540

I .460 0.967

1.964

0.978

0.544

I.455

0.968

1.964

0.978

0.666

1.333

0.982

1.963

0.987

0.500

I.500

0.964

1.964

0.975

1.976

0.545

1.454

0.968

1.964

0.978

1.975

Model

and B. Varughese

9.852 e cz

D.R.=O.I D.R.=0.2 D.R.=0.3 D.R.=0.4 D.R.=0.5 D.R.=0.6 D.R.=0.7 D.R.=0.8 D.R.=0.9 Exact

9.80-

0

0 :

‘: : :

: : lJ : = .

--*

‘. :

*. ‘. *. ‘.

=. :

9'600 II

1

2I

3I

:

--a---Y---*a--o--

D.R.=O.l D.R.=0.2 D.R.=0.3 D.R.=0.4 D.R.=0.5 D.R.=0.6 D.R.=0.7 D.R.=0.8 D.R.=0.9 D.R.=I .O & Exact

*.

4II

5

6I

7I

8I

9I

Number of elements

Fig. 8. Stress values before the drop-off for various ratios

(example

2).

y.7L-----0 12

drop

3

4

5

6

7

a

9

Number of elements Fig. 9. Stress values

In all the three meshings, the sizes of the elements have been varied so that the drop-off point passes through the different zones of the element (Tables 1 and 2). The drop ratio has been kept constant as 0.5. --o---o---LI---c-

9.75-

before the drop-off ratios (example 3).

for various

drop

Table 1 shows the values of the non-dimensional axial deformation (d) at the location of the drop (A) and at the tip of the beam (B), which are computed as A,E,G,/PL and A,E,&/PL respectively. Here A, is the area of cross-section of the beam at the thick section, EL is the modulus of elasticity in the fiber direction for examples 2 and 3 (E, = E for example 1) and 6, and S, are the deformations at A and B respectively. In the first instance, i.e. when the drop-off coincides with one end node of the elements, all the mesh types produce exact results. Subsequently, as the element sizes are altered, the drop-off passes through different locations of the element. As a result the change in thickness is not represented exactly and the thicker section of the beam gets smeared into the thinner section. Therefore, the stiffness at the thicker section is underestimated and that at the thinner section is overestimated. It can be seen that, in general, the three-noded elements (mesh type 3) perform better in prediction of displacements in both A and B than the two-noded element models. For mesh type 3, as the interior nodes are shifted. the normalized deformations diverge only gradually from the exact solution until the beam is modeled with a single three-noded element. In the case of mesh type 2, the divergence is initially faster and then converges to the exact solution. This convergence is because the common end node of the two elements is reached at the drop and thereby smearing is avoided. However, it may be noted that when the beam is modeled with a single two-noded element, the smearing is maximum and hence the results are further away from the exact solution. The deformation values for the composite beams (examples 2 and 3) are also listed in Table 1. At the very outset. it is visible that the values at the tip (under B) do not vary appreciably for mesh types 2 and 3. This shows that the proposed formulation can

Drop-off in composite structure analysis

869

Table 2. Stress before (A) and after (B) the drop-off for various meshing Composite 45/o/45

Isotropic A

B

Model

A

Composite 90/o/90 B

outer core

B

A outer core

5.000

10.000

0.359

9.640

10.000

0.244

9.756

10.000

5.000

10.000

0.359

9.640

10.000

0.244

9.756

10.000

5.000

10.000

0.359

9.640

10.000

0.244

9.756

10.000

5.545

9.090

0.362

9.698

9.940

0.245

9.792

9.960

5.399

9.200

0.362

9.688

9.950

0.245

9.788

9.967

5.323

9.350

0.361

9.672

9.968

0.244

9.766

9.976

5.833

8.333

0.364

9.744

9.893

0.246

9.824

9.930

5.600

8.800

0.362

9.704

9.936

0.245

9.798

9.958

5.421

9.145

0.361

9.680

9.960

0.245

9.784

9.968

6.150

7.693

0.365

9.776

9.859

0.246

9.848

9.906

5.600

8.800

0.362

3.696

9.942

0.245

9.793

9.962

5.444

9.112

0.361

9.680

9.960

0.245

9.784

9.968

6.428

7.143

0.366

9.800

9.834

0.247

9.864

9.889

5.400

9.200

0.361

9.672

9.967

0.244

9.776

9.979

5.445

9.110

0.361

9.682

9.958

0.245

9.784

9.968

6.666

6.661

0.366

9.816

9.818

0.247

9.872

9.880

5.000

10.000

0.359

9.640

10.000

0.244

9.756

10.000

5.455

9.091

0.361

9.684

9.956

0.245

9.784

9.968

accommodate the drop-off of composite layers efficiently. Conversely, the results under A (at the drop-off) are varying, but only slightly in mesh type 1 and the agreement is very good in mesh types 2 and 3. This close agreement is due to the high orthotropy (EL/ET.= 40) considered in these examples. In these examples the 45” or the 90” ply has been dropped off. The contribution of these layers in stiffness is low, due to their considerably low elastic modulus. However, it may be noted that the present model is not capable of accommodating the shear-extension coupling which occurs in the case of 45/O/45 lay-up. To include this effect, a more detailed model including the out-of-plane displacement (a) is required. It may be noted that, in the present examples, the deformation varies linearly along the length. Therefore, the linear shape functions of the two-noded element is enough to model the beam. In Table 2, the stresses before (under A) and after (under B) the drop-off are tabulated. The stress before the drop-off is computed as 6,E/L and that after the drop-off is computed as (S, - 6,)E/ (L - L, ), where E represents the modulus of elasticity of the material along the beam axis. For composite

beams, the stresses are listed separately for outer layers and core layer. The stresses before and after the drop-off increase and decrease respectively as the interior nodes are shifted in all the mesh types in example 1; this is again due to the smearing action across the drop in the beam. The nature of variation of values is similar to that in the case of normalized deformations. Here too, mesh type 3 produces consistently better results compared to the other meshes, as the element sizes are varied. Except for type 1, the variation of stress values is relatively insignificant in the case of examples 2 and 3, compared to that in the case of the isotropic beam. Among the two composite beams, the latter one shows the least variation because of the still higher orthotropy of its outer layers. 4.2. Effect of changing drop ratio on the response In this section, the effect of the location of the drop on the response of the beam has been studied. The cantilever beam (Fig. 3) has been analysed for axial loading for varying drop ratio (D.R.). The drop ratio, which is the ratio of L, to L, has been varied from 0. I to 1.0 by increasing the lengths of outer layers (L,)

A. Mukherjee

870

of the beam keeping the span (L ) constant. The span (L) and the load(P) are kept the same as 10.0 and 5.0 respectively. For each drop ratio, the analysis has been carried out using meshes of single, two, four and eight three-noded elements. The nodal distances are always kept the same. Thus, for any mesh there are chances that the drop may or may not occur at the nodal points. Figures 4-6 show the non-dimensionalized axial deformation (d) at the tip of the beam against number of elements. The finite element solutions are plotted, along with the theoretical values for various drop ratios. Referring to Fig. 4 (example I), it can be observed that the finite element results tend to reach the exact solution as the number of elements increases. The analysis at drop ratio 0.5 can be considered as a special case where all the meshes, except the single element mesh, give exact solutions. It is to be noted that for the single element case, the drop occurs at the mid-side node, which is not adequate to produce an exact solution. In general, it is seen that the difference between exact and finite element solutions, due to drop-off, is minimized as the number of elements are increased. It has been computed that there is a difference of2.63.6% between the single element result and the exact solution, when the drop ratio is varied from 0.1 to 0.9. For the eight element mesh this difference is from 0.4 to 0.6% for the same range of drop ratio. Figures 5 and 6 show the values of dfor various drop ratios for the two composite beams (examples 2 and 3 respectively). It has been found that all the finite element solutions coincide with the exact solutions for all drop ratios in the case of example 3. This is due to the high orthotropy considered in the analysis. In example 2, also, this can be noticed, except at two drop ratios (0.2 and 0.7) where the single element mesh values do not coincide with exact values. Figures 7-9 show the stress values in the core layer (before the drop-off) for examples I, 2 and 3 respectively. In all the examples, it can be seen that the stresses computed from displacements of FEM gradually converge to the exact solution as the number of elements increases. It may be noted that the stress values are exact whenever the drop coincides with one common end node. In all the other cases, the stress values come closer to the exact solution as the drop ratio is increased. In order to know the effect of drop ratio and number of elements in these examples, the percentage variation of stress values has been computed. For drop ratio 0.1, the eight element mesh result gives a difference of 7%, while the single element produces a difference of 49% with the values from the exact solution in example I. These differences are computed to be 0.37 and 2.4% in example 2 and 0.25 and I .6% in example 3. 5.

structures has been of laminated composite presented. Since the drop-off is spread over a refinite elements are not gion, the conventional adequate to accommodate the drop-off effectively. The present element is able to take care of the drop-off anywhere within the beam. Hence the mesh division can be made independent of the drop location and the drop-off need not pass through the nodal points. Studies have been conducted to observe the performance of the present element for the above features. Numerical results from FEM on isotropic and composite layered beams, subjected to membrane loading, have been compared with theoretical solutions and have been found to be encouraging. 2. In the present formulation, the stiffness matrix has to be developed at first for each layer in an element and then superimposed to get the stiffness matrix for the element; this is because of the differences in lengths of various layers. The Gauss points for each layer have been found from a linear relationship between the geometric and 5 coordinate. In this study, where only straight beams are discussed, this relation remains linear for both three-noded and two-noded elements and it may be noted that the linearity is not affected by the degree of the shape functions. It is also to be noted that the location of the drop within a two-noded element does not affect the stiffness of the element, since the straindisplacement matrix carries only constant terms due to the linear shape functions. 3. The examples are discussed under two sections, namely effect of mesh size (keeping the drop ratio constant) and effect of changing drop ratio. In the first section, two-noded, as well as threenoded elements have been employed. It has been noticed that the mesh type 3 (with two three-noded elements) consistently produced results in closer agreement with exact solutions as the element sizes were varied. The other mesh types (with twonoded elements) showed agreement when one of the end node is at or very near the drop-off. In the second section, the effect of varying drop ratio using a different number of three-noded elements has been presented. The performance is improved as the number of elements increases; this is especially evident in the case of stresses. The high orthotropy of the material of the laminated composite beams has greatly reduced the significance of drop-offs in them. REFERENCES 1.

B. L. Kemp and E. R. Johnson, of a graphite epoxy internal plies. Proc. Structural,

CONCLUSIONS

I. The formulation of a beam element inclusion of ply drop-off in the global

and B. Varughese

for the analysis

Srructurul

laminate

Response and failure containing terminating

AIAA/ASME,!ASCE/AHS Dynumic.s and Muteriul

26th

Conf:.

AIAA, Florida. April (1985). 2. J. M. Curry, E. R. Johnson and J. H. Starnes Jr, Effect of dropped plies on the strength of graphite epoxy laminates.

AIAA

J. 30, 449456

(1992).